Isabel K Darcy Mathematics Department Applied Mathematical and
Isabel K. Darcy Mathematics Department Applied Mathematical and Computational Sciences (AMCS) University of Iowa http: //www. math. uiowa. edu/~idarcy This work was partially supported by the Joint DMS/NIGMS Initiative to Support Research in the Area of Mathematical Biology (NSF 0800285). © 2008 I. K. Darcy. All rights reserved
Rob Scharein’s Knot. Plot. com
http: //updatecenter. britannica. com/art? assembly. Id=91&type=A
Duplex DNA knots produced by Escherichia coli topoisomerase I. Dean FB, Stasiak A, Koller T, Cozzarelli NR. , J Biol Chem. 1985 Apr 25; 260(8): 4975 -83.
Intricate Knots in Proteins: Function and Evolution Peter Virnau, Leonid A. Mirny, and Mehran Kardar, PLo. S Comput Biol. 2006 September; 2(9): e 122.
Statistics of knots, geometry of conformations, and evolution of proteins. Rhonald C. Lua, Alexander Y. Grosberg PLo. S Comput Biol. 2006 May; 2(5) unknot 3. 1 4516 164 9 3 Center 4692 20 3 1 Random 4697 15 0 1 Direct 5. 2 and more complicated knots for random closure
Mathematical Model Protein = DNA = =
http: //www. accessexcellence. org/R C/VL/GG/images/dna_replicating. g if http: //www. ch. cam. ac. uk/magnus/ molecules/nucleic/dna 1. jpg
(J. Mann) http: //www. sbs. utexas. edu/herrin/bio 344/ Postow L. et. al. PNAS; 2001; 98: 8219 -8226
Topoisomerase II performing a crossing change on DNA: Cellular roles of DNA topoisomerases: a molecular perspective, James C. Wang, Nature Reviews Molecular Cell Biology 3, 430440 (June 2002)
Mathematical Model Protein = DNA = =
C. Ernst, D. W. Sumners, A calculus for rational tangles: applications to DNA recombination, Math. Proc. Camb. Phil. Soc. 108 (1990), 489 -515. protein = three dimensional ball protein-bound DNA = strings. Protein-DNA complex Heichman and Johnson Slide (modified) from
Many mathematicians solve equations x+3=5 x = 2 is a solution: 2 + 3 = 5 x = 1 is not a solution: 1 + 3 = 4 = 5
Solving tangle equations Equation A is from: Path of DNA within the Mu Transpososome Transposase Interactions Bridging Two Mu Ends and the Enhancer Trap Five DNA Supercoils, Cell 2002 Shailja Pathania, Makkuni Jayaram and Rasika Harshey
Old Pathway: Path of DNA within the Mu Transpososome: Transposase Interactions Bridging Two Mu Ends and the Enhancer Trap Five DNA Supercoils. Shailja Pathania, Makkuni Jayaram and Rasika M Harshey Cell 2002 New Pathway (post tangle analysis): Interactions of Phage Mu Enhancer and Termini that Specify the Assembly of a Topologically Unique Interwrapped Transpososome. Zhiqi Yin, Asaka Suzuki, Zheng Lou, Makkuni Jayaram and Rasika M. Harshey
Solving tangle equations Equation A is from: Path of DNA within the Mu Transpososome Transposase Interactions Bridging Two Mu Ends and the Enhancer Trap Five DNA Supercoils, Cell 2002 Shailja Pathania, Makkuni Jayaram and Rasika Harshey
A difference topology experiment:
x 2 + x 3 = 4
x 2 + x 3 = 4 x 1 + x 3 = 3 x 1 + x 2 = 3 implies x 1 = 1, x 2 = 2, x 3 = 2 Thus T =
Theorem [Darcy, I. K. , Luecke, J. , Vazquez, M. , Tangle analysis of difference topology experiments: applications to a Mu protein-DNA complex ]: If T is a solution to the Mu-Cre equations, and if either 1. ) T is rational or split or has parallel strands or 2. ) If T has fewer than 9 crossings, Then T = (assuming a particular handedness of the products)
Also solved up to 8 crossings computationally in Darcy, I. K. , Bhutra, A. , Chang, J. , Druivenga, N. , Mc. Kinney, C. , Medikonduri, R. K. , Mills, S. , Navarra Madsen, J. , Ponnusamy, A. , Sweet, J. , Thompson, T. , Coloring the Mu Transpososome, BMC Bioinformatics. 2006 Oct 5; 7: 435.
How to solve tangle equations 1. ) Brute force Ex: solve x + 3 = 5 0 + 3 = 5, 1 + 3 = 5, 2 + 3 = 5
How to solve tangle equations 1. ) Brute force Ex: solve x + 3 = 5 0 + 3 = 5, 1 + 3 = 5, 2 + 3 = 5 2. ) Make simplifying assumptions
How to solve tangle equations 1. ) Brute force Ex: solve x + 3 = 5 1 + 3 = 5, 2 + 3 = 5 2. ) Make simplifying assumptions 3. ) Use mathematics
A+B=C 2+0 =2 2 + -2 = 0 Most tangles don’t have inverses
Some tangles (but not all) can be classified using fractions.
Which tangles are rational?
Which tangles are rational? This one is not rational. The others are all rational
Rob Scharein’s Knot. Plot. com
Topoisomerase II performing a crossing change on DNA: Cellular roles of DNA topoisomerases: a molecular perspective, James C. Wang, Nature Reviews Molecular Cell Biology 3, 430440 (June 2002)
3 D visualization software to analyze topological outcomes of topoisomerase reactions I. K. Darcy, R. G. Scharein and A. Stasiak Nucleic Acids Research 2008 36(11): 3515 -3521
Cover: Visual presentation of knot distance metric created using the software Topo. ICE-X within Knot. Plot. A pair of knots in this graph is connected by an edge if they can be converted into one another via a single intersegmental passage. This graph shows all mathematically possible topoisomerase reaction pathways involving small crossing knots. Darcy et al. (Nucleic Acids Res. , 2008; 36: 3515– 3521).
Acknowledgements: Rob Scharein Andrzej Stasiak John Luecke Mariel Vazquez Steve Levene Bhutra, A. , Chang, J. , Druivenga, N. , Mc. Kinney, C. , Medikonduri, R. K. , Mills, S. , Navarra Madsen, J. , Ponnusamy, A. , Sweet, J. , Thompson, T.
A protein bound to two segments of DNA can be modeled by a tangle. An electron micrograph of the Flp DNA complex is shown below: Electron micrograph courtesy of Kenneth Human and Steve Levene
The tangle equations corresponding to the electron micrograph:
C. Ernst, D. W. Sumners, A calculus for rational tangles: applications to DNA recombination, Math. Proc. Camb. Phil. Soc. 108 (1990), 489 -515. protein = three dimensional ball protein-bound DNA = strings. Protein-DNA complex Heichman and Johnson
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